Dynamical systems that sort lists, diagonalize matrices and solve linear programming problems

The author establishes a number of properties associated with the dynamical system H˙=[H,[H,N]], where H and N are symmetric n-by-n matrices and [A,B]=AB-BA. The most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices. Particular emphasis is placed on the role of this equation as an analog computer. For example, it is shown how to map the data associated with a linear programming problem into H(0) and N in such a way as to have H˙=[H[H,N]] evolve to a solution of the linear programming problem. This result can be applied to find systems that solve a variety of generic combinatorial optimization problems, and it also provides an algorithm for diagonalizing symmetric matrices